#### Vol. 11, No. 7, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
On $s$-harmonic functions on cones

### Susanna Terracini, Giorgio Tortone and Stefano Vita

Vol. 11 (2018), No. 7, 1653–1691
##### Abstract

We deal with nonnegative functions satisfying

where $s\in \left(0,1\right)$ and $C$ is a given cone on ${ℝ}^{n}$ with vertex at zero. We consider the case when $s$ approaches $1$, wondering whether solutions of the problem do converge to harmonic functions in the same cone or not. Surprisingly, the answer will depend on the opening of the cone through an auxiliary eigenvalue problem on the upper half-sphere. These conic functions are involved in the study of the nodal regions in the case of optimal partitions and other free boundary problems and play a crucial role in the extension of the Alt–Caffarelli–Friedman monotonicity formula to the case of fractional diffusions.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/apde

We have not been able to recognize your IP address 34.207.230.188 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.